Weak-Strong Uniqueness Principle for Hamilton-Jacobi Equations

TL;DR

This paper proves a weak-strong uniqueness principle for Hamilton-Jacobi equations, establishing that differentiable solutions with Lipschitz gradients are unique among semi-concave weak solutions without requiring convexity assumptions.

Contribution

It introduces a new uniqueness result for Hamilton-Jacobi equations that does not depend on convexity or variational representations.

Findings

Differentiable solutions with Lipschitz gradients are unique among semi-concave weak solutions.

The result applies without convexity or concavity assumptions on initial data or nonlinearity.

The theorem broadens the scope of uniqueness results for Hamilton-Jacobi equations.

Abstract

We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on the initial condition or the nonlinearity, and can therefore be utilized in contexts where the viscosity solution admits no standard variational representation.